Psychology 320: Psychological Statistics
Professor: Howard B. Lee
Lecture Notes
Week 8 : Chapter 8
Professor: Howard B. Lee
Lecture Notes
Week 8 : Chapter 8
Review of sampling distribution
One sampling distribution is one of sample means. From the Central Limit Theorem
we know that regardless of the shape of the population distribution, if all samples of
size n are drawn from the population, the sample means
will be approximately normal in their distribution.
Also the mean of the sample means is equal to the population mean.
Lecture 18
Student 's T-Test
Ex. Is therapy A better than therapy
B?
Is teaching method 1 better than teaching method 2?
Is the lot acceptable ( in quality control ) ?
Ex. Manufacturer of televisions warranty is 3 years (population mean is
3 years).
7 owners found their set to fail before 3 years.
The sample = the number of years until the set failed: 2.5, 1.9, 1.7, 2.8,
2.9, 2.7., 2.1
Question:
1) Is this a chance occurrence - meaning there is nothing wrong
with the manufacturing process?
2) Does the manufacturer have production
problems?
Pre-established information, such as a manufacturer's claim,
should be taken as the null hypothesis. * In this class, the null hypothesis is the only one that will contain
an equal sign. Steps to hypothesis testing: 3) Student's t-test ( use N to calculate SD on your calculator, not n-1! )
-3.529 = "the test statistic"
Use the t-table on p. 405
ho: manufacturer's claim is
correct.
h1: manufacturer's claim is incorrect/wrong.
M = 2.371
SD = .4366
Lecture 19
4) The Decision rule:
"5% of the time I will reject ho when ho is true."
Reject ho if the test statistic is less than the value given on p. 405.
With the table, look under one-tailed values, .05 h1 tells whether it is a one-tailed (directional) or a two-tailed
( non-directional ) test. The coin is not fair. Use table p. 405 for decision rule. Anything less than the value found in table you reject ho.
Any value greater than t (critical value) or less than -t(critical value)
reject ho.
T critical value found in Table on Page 405.
Student t-value Table from page 405 of text. Ex. Given: Your decision will be: The correct answer is choice "a".
Is Alpha = .05 that good?
There is no scientific basis for using Alpha =.05 or .01.
Decision = reject ho 5) Conclusion = There is evidence
that the manufacturer's claim is wrong. T-tests
1. one sample vs. population
Two types of teaching methods where you want to know which one is better.
3) Correlation Independent groups
The two groups are not related or connected in any meaningful way.
Dependent groups
The two groups are related. Ex. Two different word processors with five support staff members.
ho: u1 = u2
The test statistic for 2 independent group test: Decision rule:
Look at h1 to determine if it is
a one or two tailed tests.
Combine this with the h1 and the table on p. 405.
Note: in some places, the symbol "u" is used to designate the population mean.
ho: u = 3
h1: u < 3
t = -3.53
degrees of freedom (df) = n -1 or 7 -1 = 6.
on top of page; probability alpha values are for two-tailed
tests.
h1: u < 3 = one tailed (left)
h1: u < 3 one-tailed ( left )
h1: u > 36 one-tailed ( right )
Stated in the problem is some kind of specific direction.
Ex. Is there a problem with the TV manufacturer's claim?
"TVs fail before 3 years" tells some kind of direction.
"Left-tailed test"
degrees of freedom = n - 1 where n = sample size.
df = n-1 = 7-1 = 6
Italized value is the critical value sought.
alpha .50 .10 .05 .02 .01 two-tailed
.25 .05 .025 .01 .005 one-tailed
df - - - - - -
1 1.00 6.34 12.71 31.82 63.66 -
2 .816 2.92 4.30 6.96 9.92 -
3 .765 2.35 3.18 4.54 5.84 -
4 .741 2.13 2.78 3.75 4.60 -
5 .727 2.02 2.57 3.36 4.03 -
6 .718 1.94 2.45 3.14 3.71 -
ho: u > or = 3
h1: u < 3
t = -3.53
n = 7
alpha = .05
a) reject ho
b) do not reject ho
c) accept ho
d) reject h1
e) none of these or not enough information
Stay away from "c" and "d".
Choice "c", accept ho, could only happen
if you know the probability of type II error!
All you have to do is look at the table and find df.
* For left-tailed tests look at h1 and use < as an arrowhead pointing to
the left.
* You must change the sign
for left-tailed tests.
If Alpha = .05 and Beta = .25, you could possibly accept ho. However without knowledge of Beta, the best
one can do is either reject ho or do not reject ho.
It depends on the case.
It is a convention used in psychology where if you get it a paper is
publishable.
2. difference between two samples ( 2 versions ):
Data Sample 1 Data Sample 2
Treatment A Treatment B
Two types of drug therapies.
Sex differences, males compared to females.
The 4 people in group 1 are different from the 4 people in group 2.
The total number of people is 8.
One person in group 1 is matched
or paired with a person in group 2.
You can use the same person in group 1 and group 2.
ho: u1 = u2
h1: u1 is not = u2 ( a two-tailed test )
* You can't tell if they are independent or dependent groups by looking
at ho and h1.
ho: u1 = u2
h1: u1 > u2
Where u1 - u2 = 0 and u1 - u2 > 0
This is a one-tailed test ( right ).
h1: u1 < u2
Where u1 - u2 = 0 and u1 - u2 < 0
This is a one-tailed test ( left ).
Using table p. 405, find the critical value.
If ho is rejected the conclusion would be something like; " There is
sufficient evidence that treatment A is better than treatment B".
Lecture 20
Two independent samples
T-test: Example
The two groups are similar to begin with, give different treatment, take
sample, measure.
Need alpha, two tailed test, degreees of freedom=n1 + n2 - 2
Decision rule: reject ho if test statistic is > 2.31 or
<-2.31, otherwise do not reject ho.
Experiment between two groups of students. Measure and compare students
who only attend lecture and students who attend lecture and read their books.Lecture Only Read and Lecture
15,20,13,14,25
26,22,35,18,33 
4) Decision Rule
For this problem alpha = .05, df = 5+5-2 = 8
look at Table C on Page 405 of text.
For 8 df, .05 two-tailed, the critical value is 2.31.
Decision: Since t = -240 and -2.40 is less than -2.31, reject ho.
Some previous test items...
A health educator wants to evaluate the effect of a dental film on the
frequency with which children brush their teeth. Eight randomly chosen kids
are measured on the number of times they brushed their teeth for one month
before viewing the dental film and again for a month after the film. The data
are given below:
Kids 1 2 3 4 5 6 7 8 Before 25 28 22 30 26 24 25 22 After 28 29 25 30 25 28 28 241. The null hypothesis would be:
2. The value of the test statistic is:
a) 3.074
b) 3.071
c) 3.069
d) 2.873
e) 2.871
3. The critical value if alpha = .05 is:
a) +2.36
b) + or - 2.36
c) + or - 1.90
d) -2.36
e) - 1.90
4. The appropriate decision is:
a) reject ho
b) retain ho
c) accept ho
d) do not reject ho
e) b and c
5. The coefficient of determination is 0.217 based on eighteen pairs
of scores. The corresponding correlation coefficient is:
a) significant at the .05 level, one-tailed
b) significant at the .05 level, two-tailed
c) significant at the .01 level, one-tailed
d) a and b
e) not statistically significant
6. The mean in the t-distribution equals:
a) M
b) u
c) 0
d) 1
e) r
7. A research psychologist found one sample mean of 13 based on a
n of 12. The second sample mean was 18 based on a n of 10.
The design of this study should be analyzed with:
a) independent samples t-test
b) dependent samples t-test
c) one sample t-test
d) a or b
e) none of these
8. According to the textbook, the most cost effective design analyzed
by the dependent samples t-test is:
a) matched subjects
b) independent samples
c) within-subjects
d) random samples
e) experimental and control group
